We study the distribution of the traces of the Frobenius endomorphism of genus g curves which are quartic non-cyclic covers of P 1 Fq , as the curve varies in an irreducible component of the moduli space. We show that for q fixed, the limiting distribution of the trace of Frobenius equals the sum of q + 1 independent random discrete variables. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution. Finally, we extend these computations to the general case of arbitrary covers of P 1 Fq with Galois group isomorphic to r copies of Z 2Z. For r = 1, we recover the already known hyperelliptic case.

We give an explicit description of fundamental domains associated with the p-adic uniformisation of families of Shimura curves of discriminant Dp and level N ≥ 1, for which the one-sided ideal class number h(D, N ) is 1. The results obtained generalise those in [GvdP80, Ch. IX] for Shimura curves of discriminant 2p and level N = 1. The method we present here enables us to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, p-adic good fundamental domains, and their stable reductiongraphs. The method is based on a detailed study of the modular arithmetic of an Eichler order of level N inside the definite quaternion algebra of discriminant D, for which we generalise the classical results of Hurwitz [Hur96]. As an application, we prove general formulas for the reduction-graphs with lengths at p of the families of Shimura curves considered.

No abstract

No abstract

We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion Q-algebra, is in bijection with the set of certain classes of p-adic binary quadratic forms, where p is a prime dividing the discriminant of the quaternion algebra. The classes of p-adic binary quadratic forms are obtained by the action of a discrete and cocompact subgroup of PGL 2 (Q p ) arising from the p-adic uniformization of the Shimura curve. We finally compute families of p-adic binary quadratic forms associated to an infinite family of Shimura curves studied in [2]. This extends results of Alsina-Bayer [1] to the p-adic context.